Math 555 Differential Equations I

Unit III Exam: Review Guide

1. Determine the Taylor series for the given function centered about the given point.

	a.)	\(\sin x\),   \(x_0 = 0\)
	b.)	\(x^2\),   \(x_0 = -1\)
	c.)	\(\dfrac{1}{1 + x}\),   \(x_0 = 2\)

2. Determine the radius of convergence of the power series \(\displaystyle \sum_{n=1}^\infty \dfrac{n!x^n}{n^n}\).

3. You wish to find a power series solution to the SODE \(y'' - xy' - y = 0 \) centered around the point \(x_0 = 1\). What is the recurrence relation?

4. Find the first four terms of each of the power series solutions to the initial value problem. \[\begin{cases} (1-x)y'' + xy' - y = 0, \\ y(0) = -3, \\ y'(0) = 2. \end{cases}\]
5. Determine a lower bound for the radius of convergence of series solutions to the SODE about each given value of \(x_0\) without solving the equation. \[\begin{cases} (x^2 - 2x - 3) y'' + xy' + 4y = 0; \\ x_0^1 = 4,\ \ x_0^2 = -4,\ \ x_0^3 = 0. \end{cases}\]
6. Determine the values of \(\varphi''(x_0)\), \(\varphi'''(x_0)\), \(\varphi^{(4)}(x_0)\), and \(\varphi^{(5)}(x_0)\) if \(y = \varphi(x)\) is a solution of the IVP. \[\begin{cases} x^2y'' + (1 + x)y' + (3\ln x)y = 0, \\ y(1) = 2, \\ y'(1) = 0. \end{cases}\]
7. Find all singular points of the SODE and determine whether or not they are regular. \[ (x\sin x)y'' + 3y' + xy = 0 \]
8. Use the definition of the Laplace transform to compute \(\mathcal{L}\{f(t)\}\) for the given functions.

	a.)	\(t^n\), where \(n\) is a positive integer
	b.)	\(\sin t\)
	c.)	\( \sinh t\)

9. Find the inverse Laplace transform \(\mathcal{L}^{-1}\{F(s)\}\) of the following functions.

	a.)	\( \dfrac{2s+1}{s^2 - 2s + 2} \)
	b.)	\( \dfrac{2s-3}{s^2 + 4s - 4} \)

10. Use the method of Laplace transforms to solve the initial value problem. \[\begin{cases} y'' - y' - 6y = 0, \\ y(0) = 1, \\ y'(0) = -1. \end{cases}\]
11. Use the method of Laplace transforms to solve the initial value problem. \[\begin{cases} y^{(4)} - 4y = 0, \\ y(0) = 1,\ \ y'(0) = 0,\ \ y''(0) = -2,\ \ y'''(0) = 0. \end{cases}\]
12. Sketch the graph of the function and rewrite it in terms of the unit step functions \(u_c(t)\), then find the Laplace transform of \(f\). \[f(t) = \begin{cases} \phantom{-}1, & 0 \leq t < 1, \\ -1, & 1 \leq t < 2, \\ \phantom{-}1, & 2 \leq t < 3, \\ -1 & 3 \leq t < 4, \\ \phantom{-}0, & t \geq 4. \end{cases}\]
13. Sketch the graph of the function, write it as a piecewise function, and compute its Laplace transform. \[ f(t) = u_1(t) + 2u_3(t) - 6u_4(t) \]
14. Find the inverse Laplace transform \(\mathcal{L}^{-1}\{F(s)\}\) of the function \[F(s) = \dfrac{e^{-s} + e^{-2s} - e^{-3s} - e^{-4s}}{s}.\]
15. Find the solution of the initial value problem, \[\begin{cases} y'' + y = f(t), \\ y(0) = 0, \\ y'(0) = 1, \end{cases}\] where \[f(t) = \begin{cases} \tfrac{t}{2}, & 0 \leq t < 6, \\ 3, & t \geq 6. \end{cases}\]

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